Glossary

7.3 Normal and Oblique Shock Waves

4 min read•august 16, 2024

Normal and oblique shock waves are crucial phenomena in supersonic flows. These discontinuities cause abrupt changes in flow properties, playing a key role in compressible fluid dynamics and aerospace engineering.

Understanding shock waves is essential for designing supersonic aircraft, rocket nozzles, and wind tunnels. We'll explore their formation, characteristics, and the equations governing their behavior, connecting theory to real-world applications in high-speed fluid flow.

Normal Shock Wave Formation

Characteristics and Physics of Normal Shock Waves

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  • Normal shock waves form discontinuities in flow properties occurring in supersonic flows when flow decelerates to subsonic speeds
  • Accumulation of pressure disturbances unable to propagate upstream in causes formation
  • Normal shock waves create abrupt changes in flow properties (pressure, temperature, density, velocity)
  • Thickness of normal shock wave measures extremely small (few mean free paths of gas molecules)
  • Flow properties across normal shock wave experience following changes:
    • Increase in pressure, temperature, and density
    • Decrease in flow velocity
  • transitions from supersonic (M > 1) upstream to subsonic (M < 1) downstream
  • Entropy increases across normal shock wave indicating irreversible process

Examples and Applications

  • Normal shock waves observed in supersonic wind tunnels (test section)
  • Occur in supersonic nozzles operating at off-design conditions
  • Found in front of blunt objects in supersonic flow (aircraft nosecones)
  • Appear in overexpanded rocket nozzles

Rankine-Hugoniot Equations for Normal Shocks

Fundamental Equations and Relationships

  • Rankine-Hugoniot equations consist of conservation equations relating flow properties across normal shock wave
  • Equations incorporate , momentum, and energy
  • Express Rankine-Hugoniot equations using upstream Mach number and ratio of specific heats of gas
  • Key relationships derived from equations include ratios across shock wave:
    • Pressure ratio: p2p1=1+2γγ+1(M12−1)\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1}(M_1^2 - 1)
    • Temperature ratio: T2T1=[2γM12−(γ−1)][2γ+1]M12\frac{T_2}{T_1} = \frac{[2\gamma M_1^2 - (\gamma-1)][\frac{2}{\gamma+1}]}{M_1^2}
    • Density ratio: ρ2ρ1=(γ+1)M122+(γ−1)M12\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}
    • Velocity ratio: u2u1=2+(γ−1)M12(γ+1)M12\frac{u_2}{u_1} = \frac{2+(\gamma-1)M_1^2}{(\gamma+1)M_1^2}
  • Normal shock function relates upstream and downstream Mach numbers: M22=1+γ−12M12γM12−γ−12M_2^2 = \frac{1 + \frac{\gamma-1}{2}M_1^2}{\gamma M_1^2 - \frac{\gamma-1}{2}}

Practical Applications and Problem-Solving

  • Normal shock tables based on Rankine-Hugoniot equations provide quick determination of flow properties
  • Apply equations to find unknown flow properties given known conditions before or after shock wave
  • Solve problems involving normal shocks in various engineering scenarios:
    • Supersonic inlet design for jet engines
    • Shock tube experiments in laboratory settings
    • Analysis of flow in supersonic wind tunnels

Oblique Shock Wave Concept

Formation and Characteristics

  • Oblique shock waves form at angle to flow direction when supersonic flow encounters deflection or compression
  • Occur in supersonic flow over wedges, cones, or surfaces causing flow direction change
  • Characterized by shock angle (β) and deflection angle (θ) related to upstream Mach number
  • Oblique depends on upstream Mach number and flow deflection angle
  • Can be attached (to surface) or detached (bow shock) based on flow conditions and geometry
  • Two possible oblique shock solutions for given upstream Mach number and deflection angle:
    • Weak shock (smaller β, supersonic flow behind shock)
    • Strong shock (larger β, behind shock)

Types and Examples

  • Attached oblique shocks form on sharp leading edges of supersonic aircraft wings
  • Detached bow shocks appear in front of blunt bodies in supersonic flow (spacecraft reentry)
  • Multiple oblique shocks occur in supersonic inlets of ramjet engines
  • Shock diamonds in overexpanded jet exhaust consist of series of oblique shocks and expansion fans

Oblique Shock Wave Properties

Oblique Shock Relations and Analysis

  • Derive oblique shock relations from Rankine-Hugoniot equations modified for oblique nature
  • θ-β-M relation connects deflection angle, shock angle, and upstream Mach number: tan⁡θ=2cot⁡βM12sin⁡2β−1M12(γ+cos⁡2β)+2\tan\theta = 2\cot\beta\frac{M_1^2\sin^2\beta - 1}{M_1^2(\gamma + \cos2\beta) + 2}
  • Property ratios across oblique shock depend on normal component of upstream Mach number (Mn1)
  • Calculate normal component of Mach number: Mn1=M1sin⁡βM_{n1} = M_1\sin\beta
  • Use Mn1 to determine oblique shock strength and flow properties
  • Prandtl-Meyer expansion fans may occur with oblique shocks when flow turns away from itself

Practical Applications and Tools

  • Oblique shock charts and tables based on relations aid in determining flow properties and shock angles
  • Analyze multiple oblique shocks in series by applying relations sequentially
  • Consider new flow conditions after each shock in multi-shock systems
  • Applications of oblique shock analysis:
    • Design of supersonic aircraft inlets
    • Optimization of hypersonic vehicle shapes
    • Analysis of shock wave interactions in scramjet engines

Key Terms to Review (17)

Aerodynamic drag: Aerodynamic drag is the resistance experienced by an object moving through a fluid, such as air, which acts in the opposite direction to the object's motion. This force plays a critical role in shaping the design and performance of vehicles, aircraft, and other objects, as it affects their speed, fuel efficiency, and stability. Understanding aerodynamic drag is essential for optimizing performance and minimizing energy loss during high-speed travel.
Compressible Flow: Compressible flow refers to the fluid dynamics where the density of the fluid changes significantly in response to pressure variations, commonly occurring at high velocities. In such flow regimes, factors like the speed of sound and Mach number become crucial, as they help characterize how the flow behaves, especially when shock waves are present. Understanding compressible flow is essential for analyzing phenomena such as shock waves and the behavior of gases moving at speeds approaching or exceeding that of sound.
Computational Fluid Dynamics: Computational Fluid Dynamics (CFD) is a branch of fluid mechanics that utilizes numerical analysis and algorithms to solve and analyze problems involving fluid flows. By applying computational methods, CFD allows for the simulation of complex fluid behavior in various contexts, helping to predict how fluids interact with surfaces, how shock waves form, and how flow can be controlled. This powerful tool is essential for understanding the dynamics of fluids in engineering applications, environmental studies, and aerodynamics.
Conservation of Mass: Conservation of mass is a fundamental principle stating that mass cannot be created or destroyed in a closed system, meaning the total mass of the system remains constant over time. This principle is crucial in fluid dynamics as it underpins various equations and concepts related to the flow and behavior of fluids.
Conservation of Momentum: Conservation of momentum is a fundamental principle stating that the total momentum of a closed system remains constant over time, as long as no external forces are acting on it. This principle is essential for understanding how fluids behave in various scenarios, such as shock waves and vortex dynamics, and connects to the underlying equations that govern fluid motion and stress relationships.
Density Change: Density change refers to the variation in mass per unit volume of a fluid as it undergoes different physical conditions, such as pressure and temperature changes. In the context of fluid dynamics, especially in compressible flow situations like shock waves, density change is crucial in determining how the fluid behaves under rapid pressure variations. It is particularly significant during events like normal and oblique shocks, where fluids can experience sudden transitions in velocity and thermodynamic properties.
Finite Volume Method: The finite volume method is a numerical technique used for solving partial differential equations that arise in fluid dynamics by dividing the computational domain into small control volumes. This method focuses on the conservation laws, ensuring that the flow of mass, momentum, and energy are accurately represented across the boundaries of these control volumes, making it especially effective for problems involving shock waves, turbulence, and complex geometries.
Inviscid Flow: Inviscid flow refers to the motion of an ideal fluid with no viscosity, meaning there are no internal frictional forces acting within the fluid. This concept is essential in fluid dynamics as it simplifies the equations governing fluid motion, making it easier to analyze phenomena like shock waves, vortex dynamics, and potential flows without the complexities introduced by viscosity.
Mach Number: Mach number is a dimensionless quantity representing the ratio of the speed of an object to the speed of sound in the surrounding medium. This concept is crucial in understanding compressible flow, as it indicates whether a flow is subsonic, transonic, supersonic, or hypersonic, influencing phenomena like shock waves and expansion waves.
Normal Shock Wave: A normal shock wave is a type of discontinuity that occurs in supersonic flows, where there is a sudden change in flow properties such as pressure, temperature, and density. This wave is characterized by its perpendicular orientation to the direction of flow and results in a significant drop in velocity as the flow transitions from supersonic to subsonic speeds. Normal shock waves play a crucial role in compressible fluid dynamics and are fundamental for understanding how shock waves behave in various scenarios.
Oblique Shock Wave: An oblique shock wave is a type of shock wave that occurs when a supersonic flow encounters a wedge or an inclined surface, resulting in a sudden change in pressure, temperature, and density of the flow. This phenomenon is essential for understanding how compressible fluids behave when they are subjected to changes in flow direction and speed, especially in the context of aerodynamic surfaces like airfoils or supersonic nozzles.
Pressure Jump: Pressure jump refers to the abrupt increase in pressure that occurs across a shock wave as a fluid transitions from a supersonic to a subsonic state. This phenomenon is crucial in understanding the behavior of shock waves, where changes in pressure, temperature, and density happen almost instantaneously, significantly affecting the flow characteristics and overall dynamics of fluid systems.
Rankine-Hugoniot Conditions: Rankine-Hugoniot Conditions are mathematical relations that describe the conservation of mass, momentum, and energy across a shock wave or discontinuity in a fluid flow. These conditions are critical for understanding how properties like pressure, density, and velocity change when a shock wave passes through a medium, making them essential in the study of compressible fluid dynamics.
Shock Strength: Shock strength refers to the intensity or magnitude of a shock wave as it propagates through a medium, often measured in terms of pressure increase or density change across the shock. Understanding shock strength is crucial as it influences the behavior of the flow field around the shock, including changes in velocity, temperature, and density of the gas. This term plays a significant role in analyzing both normal and oblique shock waves, as higher shock strengths can lead to greater alterations in flow properties and more complex shock interactions.
Shockwave Boundary Layer Interaction: Shockwave boundary layer interaction occurs when a shock wave interacts with the boundary layer of a fluid flow, leading to significant changes in the flow characteristics. This interaction can result in flow separation, changes in pressure distribution, and altered aerodynamic forces on objects such as airfoils and wings. Understanding this phenomenon is crucial in designing high-speed vehicles and optimizing aerodynamic performance.
Subsonic Flow: Subsonic flow refers to fluid motion where the flow velocity is less than the speed of sound in that fluid. This type of flow is characterized by smooth streamlines and pressure changes that are gradual, allowing for predictable behavior. Understanding subsonic flow is crucial as it lays the foundation for analyzing how fluids behave at different speeds, particularly when transitioning to supersonic conditions or encountering shock waves.
Supersonic Flow: Supersonic flow refers to the motion of a fluid when its speed exceeds the speed of sound in that medium. This phenomenon is characterized by significant changes in pressure, temperature, and density, leading to the formation of shock waves and expansion fans. Understanding supersonic flow is crucial for analyzing various aerodynamic behaviors, particularly in the context of high-speed aircraft and rockets.
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